Place the chicken, onion, bay leaf, and head of garlic in a pot. Cover with 6 cups of water. Add more if needed to submerge the chicken fully. Add 2 teaspoons salt and bring to a boil. Reduce heat and simmer, uncovered, for 45 minutes or until the chicken is completely cooked.
Using tongs, transfer the chicken to a bowl. Transfer the garlic to a small plate.
Turn the heat on the broth to medium and continue simmering until it reduces and deepens in color, about 10–15 minutes. Reduce the liquid to about 4 cups. It’s okay if you have a little more or a little less.
Strain the broth through a fine mesh sieve into a large bowl. Discard solids. Wipe out the pot and return it to the stove.
Carefully pick the chicken from the bones. Shred the chicken and set it aside. You can freeze the bones for stock if you like. You can fry the chicken skin for a crunchy garnish or discard it.
Pop the garlic cloves from the paper and mash them into a paste with a fork. Set aside.
Heat 1 tablespoon olive oil in the pot over medium heat. Add the leek, celery, fennel, carrots, and sauté for 10–15 minutes. Adjust the heat as needed to prevent them from burning. Season with salt and pepper.
Add the dry thyme, garlic paste, and yellow split peas to the pot. Toss to coat and cook for 2 minutes. Season with salt.
Pour in 4 cups of broth and shredded chicken. See note 1.
Bring to a boil, reduce heat to low, and simmer uncovered for 30–45 minutes or until the yellow split peas are very tender. See note 2.
Stir occasionally and gently mash the split peas, adding more water if the broth evaporates too quickly. Taste and add salt and pepper.
Once the stew is thick and creamy, stir in half the minced scallions and the lemon juice. Simmer for 5 minutes. Taste and adjust seasonings. Turn off the heat.
Ladle the stew into bowls. Garnish with the remaining scallions. Enjoy!
